Sharp regularity for singular obstacle problems

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arXiv:2210.09413v1 [math.AP] 17 Oct 2022

SHARP REGULARITY FOR SINGULAR OBSTACLE

PROBLEMS

DAMI ˜

AO J. ARA ´

UJO, RAFAYEL TEYMURAZYAN, AND VARDAN VOSKANYAN

Abstract. We obtain sharp local C1,α regularity of solutions for sin-

gular obstacle problems, Euler-Lagrange equation of which is given by

∆pu=γ(u−ϕ)γ−1in {u > ϕ},

for 0 < γ < 1 and p≥2. At the free boundary ∂{u > ϕ}, we prove

optimal C1,τ regularity of solutions, with τgiven explicitly in terms of

p,γand smoothness of ϕ, which is new even in the linear setting.

MSC (2020): 35B65, 35J60, 35J75, 35B33, 49Q20, 49Q05.

Keywords: Singular obstacle problems, degenerate elliptic operators,

sharp regularity, free boundary.

1. Introduction

In this paper we study minimization problems with non-diﬀerentiable

zero order dependence. More precisely, in a bounded domain Ω ⊂Rn, for a

constant γ∈(0,1), we study regularity of minimizers of

J(u) = inf

v∈KJ(v),(1.1)

where

J(v) := ZΩ|∇v|p

p+ (v−ϕ)γdx,

and

K:= nv∈W1,p(Ω); v≥ϕ, v −g∈W1,p

0(Ω)o,(1.2)

with ϕ∈C1,β(Ω), for a β∈(0,1] and g∈W1,p(Ω). The corresponding

Euler-Lagrange equation is

∆pu=γ(u−ϕ)γ−1in {u > ϕ} ∩ Ω,(1.3)

where

∆pu:= div |∇u|p−2∇u

is the standard p-Laplacian operator, 2 ≤p < +∞. Note that the right

hand side in (1.3) blows up at the free boundary points

∂{u > ϕ} ∩ Ω,

which makes it essential to understand its eﬀect on the regularity of mini-

mizers. The parameter γ, thus, measures the magnitude of singularity.

2 D.J. ARA ´

UJO, R. TEYMURAZYAN, AND V. VOSKANYAN

Problems like (1.1) are used, for example, to model the density of a certain

chemical in reaction with a porous catalyst pellet (see, for instance, [7]), and

due to their wide range of applications, were studied by many prominent

mathematicians. In the linear setting (p= 2), the extreme cases (γ= 0

and γ= 1) of (1.1) were studied in [2] and [9] with ﬂat obstacles (ϕ≡0).

The case γ= 0 is related to jets ﬂow and cavity problems, and minimizers

are known to have local Lipschitz (optimal) regularity, as is established by

Alt and Caﬀarelli in [2]. In the nonlinear setting the extreme case of γ= 0

was studied in [12], where Lipschitz regularity of minimizers is established.

These type of problems, often referred to as Bernoulli type problems, appear

in heat ﬂows, [1], electrochemical machining, [17], etc. The case of γ= 1

resembles the classical obstacle problem, and its solution, as is shown by

Brezis and Kinderlehrer in [8], is of class C1,1,p= 2. In the nonlinear

setting, p > 2, the obstacle problem was studied in [3,13]. Its unique

solution, as is shown in [3], is of class C1,α

loc at the free boundary points with

α= min β, 1

p−1.

The problem (1.1) was studied in [21] (with p= 2 and ﬂat obstacle), where,

using a minimizer preserving scaling, it was shown that minimizers are lo-

cally of the class C1,γ

2−γ, 0 < γ < 1 (see also [4] and [5], for the problem

governed by the inﬁnity Laplacian and uniformly elliptic fully nonlinear op-

erators, respectively). There are, so called, monotonicity formulas available

in the linear case, which play a crucial role in the study of the problem.

For γ∈(0,1), the nonlinear case is covered in [18], where for obstacle type

problem (with zero obstacle) is proved that minimizers are locally of the

class C1,α with

α= min σ−,γ

p−γ,(1.4)

where σ−is the H¨older regularity exponent for the gradient of p-harmonic

functions (a−stands for any b < a). Actually, for n= 2, from [6] one

concludes that α=γ

p−γ. Observe that in all the above results (except in [3])

the obstacle is assumed to be trivial, guaranteeing a vanishing gradient of

solutions at the free boundary points, which is essential in the analysis (in

[3] obstacle is assumed to be of the class C1,β, but the zero order dependence

of the functional is smooth). The methods, used to obtain those results, fail

to work in the presence of non-trivial obstacles with large gradients at the

free boundary, and a new approach is required to tackle the issue.

In this work, we prove sharp regularity for minimizers of (1.1) both locally

and at the free boundary points. More precisely, we show that minimizers

are locally of the class C1,α, where

α= min σ−,γ

p−γ,β

p−1,(1.5)

SHARP REGULARITY FOR SINGULAR OBSTACLE PROBLEMS 3

and σ > 0 is the H¨older regularity exponent of the gradient of p-harmonic

functions. Note that our result extends the local regularity (1.4) of [18]

for problems with non-trivial obstacles, which is new even for the linear

case (p= 2). Moreover, at the free boundary points we obtain optimal C1,τ

regularity for minimizers of (1.1), which, unlike local interior estimates, does

not depend on the regularity of p-harmonic functions. To be exact, we show

that minimizers of (1.1) at the free boundary are in C1,τ , where

τ= min β, γ

p−γ,(1.6)

which generalizes the optimal regularity result obtained in [21] for the linear

case and trivial obstacle. Thus, at free boundary points the interior regu-

larity result of [18] improves substantially. Indeed, as the obstacle in [18] is

assumed to be trivial, then from (1.6), we have

τ=γ

p−γ,

which is better than αfrom (1.4). Observe also, that our result extends

(continuously) the optimal regularity result of [3], from smooth lower or-

der dependence to the singular setting (Theorem 6.2). Our approach is

r1+β

r1+τ

{u > ϕ}

Figure 1. Detachment of ufrom ϕat the free boundary.

based on geometric tangential analysis and a ﬁne perturbation combined

with adjusted scaling argument. Strictly speaking, we redeem regularity by

“tangentially accessing” the information available in the “ﬂatness regime”

(for a rather comprehensive introduction to geometric tangential analysis,

we refer the reader to [23]). In the complementary case, i.e., when the gra-

dient of a solution is bounded from below at a free boundary point, we use

an “adjusted scaling” argument to ensure that in the limit we get a linear

elliptic equation without the zero order term. The idea of the adjustment is

to get rid of those terms that blow up at the limit.

The paper is organized as follows: in Section 2, we prove existence of mini-

mizers and in Section 3, establish local sharp C1,α regularity result (Theorem

4 D.J. ARA ´

UJO, R. TEYMURAZYAN, AND V. VOSKANYAN

3.1). In Section 4, we obtain optimal regularity at free boundary points for

minimizers with small gradient (Theorem 4.1). Section 5is devoted to the

adjusted scaling argument. Finally, in Section 6, we obtain sharp regularity

for minimizers with large gradient at the free boundary points (Theorem

6.1). We close the paper with two appendices, containing some auxiliary

technical results (Appendix A) and a list of several known ones (Appendix

B), that are used in the paper.

Notations and assumptions. Hereafter Br(x0) is the ball of radius r

centered at x0,Br(0) = Br, and |Br|stands for the volume of the ball

Br. When Ω = Brin (1.2), we will often use the notation Krinstead of K.

Additionally, for a given integrable function f, we denote by (f)rits average

on the ball of radius rcentered at the origin, i.e.,

(f)r:= 1

|Br|ZBr

f(x)dx.

To avoid repetition of arguments when applying the conclusions for diﬀerent

set of functions, we introduce a function H:Rn→R+, which is assumed to

be of class C2and satisfy the following structural assumptions











|∇H(ξ)| ≤ Υω(|ξ|),

|D2H(ξ)| ≤ Λω(|ξ|)

|ξ|,

ηTD2H(ξ)η≥λω(|ξ|)

|ξ||η|2,

(1.7)

with

ω(z) := κ1zp−1+κ2z, z ≥0,

where ξ,η∈Rnand Υ >0, Λ ≥λ > 0, κ1≥0, κ2≥0 are constants, and

κ1+κ2>0. We also will use the notation

G(z) := Zz

ω(ζ)dζ =κ1

p+κ2

2, z ∈R+.(1.8)

Remark 1.1. The function H(ξ) = p−1|ξ|psatisﬁes the above conditions

with κ1= 1 and κ2= 0. The classical linear version, p= 2, is recovered by

assuming κ1= 0. An alternative example of a function Hsatisfying (1.7)

is constructed in Appendix A.

2. Existence of minimizers

In this section we show that there exists at least one minimizer of (1.1).

Unlike the regular case (γ=1), which is known to have a unique minimizer

(see, for example, [3,10,13,20,22]) in the singular setting this is not

assured.

SHARP REGULARITY FOR SINGULAR OBSTACLE PROBLEMS 5

Theorem 2.1. If ϕ∈W1,p(Ω) ∩L∞(Ω),g∈W1,p(Ω) and γ∈(0,1), then

there exists a minimizer uof (1.1). Moreover,

kukL∞(Ω) ≤max kgkL∞(Ω),kϕkL∞(Ω).(2.1)

Proof. Set

m:= inf

v∈KJ(v)≥0.

If vi∈Kis a minimizing sequence, then for i≥i0,i0∈N, one has

0≤J(vi)≤m+ 1,

hence ZΩ

|∇vi|pdx ≤pJ(vi)≤p(m+ 1),

and the Poincar´e inequality yields that the sequence viis bounded in W1,p

0(Ω).

Therefore, by Rellich-Kondrachov theorem, there is a function u∈W1,p

0(Ω)

such that, up to a subsequence,

vi→uweakly in W1,p(Ω) and vi→uin Lp(Ω).

Notice that u∈K, and thus, using the lower semi-continuity of the Dirichlet

integral, we obtain

m≤J(u)≤lim inf

j→∞ J(vi) = m,

i.e., uis a minimizer of (1.1).

To see (2.1), set

uM:= min{u, M} ∈ K,

where

M:= max kgkL∞(Ω),kϕkL∞(Ω).

Since uis a minimizer,

ZΩ

|∇u|p

pdx −ZΩ

|∇uM|

pdx ≤ZΩ(uM−ϕ)γ−(u−ϕ)γdx,

and thus

0≤Z{u>M}

|∇u|p

pdx ≤ZΩ(uM−ϕ)γ−(u−ϕ)γdx ≤0,

which implies that u=uM. Therefore, −kϕkL∞(Ω) ≤ϕ≤u≤M.

3. Local C1,α regularity estimates

One of the main steps towards the optimal regularity of minimizers is

obtaining C1,α regularity for minimizers of

Jδ(v) := ZΩ

(H(∇v) + δ(v−ϕ)γ)dx (3.1)

over the set K, deﬁned by (1.2). Here δ∈[0,1], p≥2 and Hsatisﬁes (1.7).

The key step towards the regularity is the result on the decay of integral

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arXiv:2210.09413v1[math.AP]17Oct2022SHARPREGULARITYFORSINGULAROBSTACLEPROBLEMSDAMI˜AOJ.ARA´UJO,RAFAYELTEYMURAZYAN,ANDVARDANVOSKANYANAbstract.WeobtainsharplocalC1,αregularityofsolutionsforsin-gularobstacleproblems,Euler-Lagrangeequationofwhichisgivenby∆pu=γ(u−ϕ)γ−1in{u>ϕ},for0<γ<1andp≥2.Atthefreeboun...

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Sharp regularity for singular obstacle problems

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